Indian National Mathematical Olympiad is organized by Homi Bhabha Centre for Science Education. This post is dedicated for INMO 2019 Discussion. You can post your ideas here.

**Problem 1.** Let \(ABC\) be a triangle with \(\angle BAC > 90^\circ \). Let \(D\) be a point on the line segment \(BC\) and \(E\) be a point on the line \(AD\) such that \(AB\) is tangent to the circumcircle of triangle \(ACD\) at \(A\) and \(BE\) perpendicular to \(AD\). Given that \(CA=CD\) and \(AE=CE\), determine \(\angle BCA\) in degrees

**Problem 2.** Let \({A_1}{B_1}{C_1}{D_1}{E_1}\) be a regular pentagon. For \(2\leq n \leq 11\), let \({A_n}{B_n}{C_n}{D_n}{E_n}\) whose vertices are the midpoints of the sides of \({A_{n-1}}{B_{n-1} }{C_{n-1} }{D_{n-1}}{E_{n-1}}\). All the \(5\) vertices of each of the \(11\) pentagons are arbitrarily coloured red or blue. Prove that four points among these \(55\) have the same colour and form the vertices of a cyclic quadrilateral.

**Problem 3.** Let \(m\), \(n\) be distinct positive integers. Prove that \(\gcd (m,n) + \gcd (m+1,n+1) + \gcd (m+2,n+2) \leq 2|m-n|+1\). Further, determine when equality holds.

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**Problem 4. **Let \(n\) and \(M\) be positive integers such that \(M>n^{n-1}\). Prove that there are \(n\) distinct primes \(p_1\),\(\),\(\),….,\(p_n\) such that \(p_j\) divides \(M+j\) for \(1 \leq j \leq n\).

**Problem 5.** Let \(AB\) be a diameter of a circle \(\Gamma\) and let \(C\) be a point on \(\Gamma\) different from \(A\) and \(B\). let \(D\) be the foot of perpendicular from \(C\) on to \(AB\). let \(K\) be a point of the segment \(CD\) such that \(AC\) is equal to the semiperimeter of the triangle \(ADK\). Show that the excircle of triangle \(ADK\) opposite \(A\) is tangent to \(\Gamma\).

**Problem 6.** Let \(f\) be a function defined from the set \(\{(x,y) : x,y\) are real, \(xy \neq 0\}\) to the set of all positive real number such that

(i)\(f(xy,z)=f(x,z)f(y,z),\) for all \(x,y \neq 0\)

(ii) \(f(x,1-x)=1,\) for all \(x \neq 0,1\)

Prove that(i)\(f(x,x)=f(x,-x)=1,\) for all \(x \neq 0\)

(ii) \(f(x,y)f(y,x)=1,\) for all \(x,y \neq 0\)

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